Grease an 8-inch square baking dish or other 1 1/2 quart casserole and set aside.
In a mixing bowl, whisk the eggs and then add the brown sugar, baking powder, vanilla, cinnamon, and salt. Mix well–you don’t want any lumps.
Whisk in the melted butter and milk. Add the oats and stir to thoroughly combine. Transfer the mixture to the prepared baking dish.
At this point, you can cover and refrigerate overnight or as little as one hour. (This gives the oats time to absorb some of the liquid.) When ready to bake, allow the oatmeal to sit at room temperature while the oven preheats to 350 degrees F. Bake, uncovered, for 30 minutes (give or take a few depending on oven and proportions of selected baking dish) or until just set in the middle.
Serve warm with milk or yogurt, fruit, and/or other favorite oatmeal toppings. Add a drizzle of maple syrup or a sprinkle of brown sugar if more sweetness is preferred. Baked oatmeal can also be enjoyed cold or at room temperature and may be served like a muffin, with no toppings.